In order to show the optimality of \(\overline{\omega }_{2d}\)  for Problem REF , one will need to know the quantity \(P_f(\overline{\omega }_{2d},S^{d-1})\)  by locating the absolute minima of the potential of \(\overline{\omega }_{2d}\)  on the sphere \(S^{d-1}\) .
This was done earlier for the vertices of a regular \(N\) -gon inscribed in \(S^1\)  and for the vertices of a regular simplex, cross-polytope, and cube inscribed in \(S^{d-1}\)  for Riesz potential functions (\(s\ne 0\) ) and their horizontal translations, see the works by Stolarsky, Nikolov, and Rafailov [1]}, [2]}, [3]}, [4]}. For general potentials, this has been recently done in [5]} for vertices of a regular simplex. Below, we extend one of the results of [2]}, [4]} by finding the absolute minima of the potential of \(\overline{\omega }_{2d}\)  for potential functions \(g\)  that have a convex second derivative. Our proof is different from the one in [2]}, [4]} and uses polynomial interpolation and convexity of \(g^{\prime \prime }\) . Without loss of generality, we can assume in this section that \(\overline{\omega }_{2d}=\widetilde{\omega }_{2d}=\lbrace \pm {\bf e}_1,\ldots ,\pm {\bf e}_{d}\rbrace \) , where \({\bf e}_1,\ldots ,{\bf e}_d\)  is the standard basis in \(\mathbb {R}^d\) .
